Question: Solve for $x$ : $x^2 - 12x + 35 = 0$
The coefficient on the $x$ term is $-12$ and the constant term is $35$ , so we need to find two numbers that add up to $-12$ and multiply to $35$ The two numbers $-5$ and $-7$ satisfy both conditions: $ {-5} + {-7} = {-12} $ $ {-5} \times {-7} = {35} $ $(x {-5}) (x {-7}) = 0$ Since the following equation is true we know that one or both quantities must equal zero. $(x -5) (x -7) = 0$ $x - 5 = 0$ or $x - 7 = 0$ Thus, $x = 5$ and $x = 7$ are the solutions.